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A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.

%I A327033 #8 Aug 16 2019 12:35:29
%S A327033 0,1,1,1,1,-37,-211,2311,37153,-818946931,277930363757,
%T A327033 -711223555487930419,-6367871182840222481,35351107998094669831,
%U A327033 12690449182849194963361,-15116334304443206742413679091,1431925649981017658678758915153153,-19921854762028779869513196624259348280501
%N A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.
%C A327033 a(n) is an integer, as conjectured by Thomas Ordowski and proved by the author in A309132 and A326690.
%C A327033 Ordowski also conjectured that the sequence is a subsequence of A174341.
%H A327033 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%e A327033 Prime(6) = 13 and B(12) = -691/2730, so a(6) = -691/13 + 2730/13^2 = -37.
%t A327033 a[n_] := With[{p = Prime[n]}, With[{b = BernoulliB[p - 1]}, (p  Numerator[b] + Denominator[b])/p^2]];
%t A327033 Table[a[n], {n, 1, 18}]
%o A327033 (PARI) a(n) = my(p = prime(n), b = bernfrac(p-1)); numerator(b)/p + denominator(b)/p^2; \\ _Michel Marcus_, Aug 16 2019
%Y A327033 Cf. A174341, A309132, A326690.
%K A327033 sign
%O A327033 1,6
%A A327033 _Jonathan Sondow_, Aug 15 2019